Concept of Risk Adjustment

• For each patient, we estimate a predicted probability of survival, \(P(\text{Survival})\).
  
• The estimate depends on key variables:
  • Injury Severity Score (ISS)
  • Revised Trauma Score (RTS)
  • Age Index (<= 54, > 54)
  • Mechanism (Blunt vs. Penetrating)

Probability of survival

Modern trauma registries and EHRs will do this calculation for you. Blunt and penetrating injuries use different coefficients to estimate the predicted probabilities. The survival prognosis is computed based on a logistic regression equation of the form: \[ \text{Survival Probability} = \frac{1}{1 + e^{-b}} \]

where \[ b = \beta_{0} + \beta_{1} \times \text{RTS} + \beta_{2} \times \text{ISS} + \beta_{3} \times \text{AgeIndex} \]

From Individual to System-Level Benchmark

• Compute \(P(\text{Survival})\) for each patient.
  
• Sum these probabilities across all patients -> Expected Survivors.
• Sum \(1 - P(\text{Survival})\) across all patients -> Expected Decedents.
• Compare to Observed Survivors and Decedents from actual outcomes.

Why This is a Benchmarking Standard

• The ACS Trauma Quality Improvement Program (TQIP) uses similar models for national benchmarking.
  
• This approach adjusts for injury severity, physiology, and demographics.

Why This is a Benchmarking Standard

• It provides a fair, risk-adjusted performance measure.
  
• Centers can identify opportunities for improvement based on deviation from expected outcomes.
  > ACS Trauma Quality Programs
  > Trauma Risk Adjustment Overview (PubMed Central)

Step 1: Compute observed survivors

\[ A = n - n_{\text{deaths}} \]

\[ A = 900 - 40 = 860 \]

Step 2: Define expected survivors

\[ B = \sum P(Survival) = 750.3638 \]

Step 3: Apply W-score formula

\[ W = \frac{A - B}{C} \times 100 \]

Substitute known values:

\[ W = \frac{860 - 750.3638}{900} \times 100 \]

Step 4: Compute W-score

\[ W = \frac{109.6362}{900} \times 100 = 12.18 \]

Step 5: Inference

• \(W = 12.18\)
  -> The center achieved about 12 more survivors per 100 patients than expected.
  
• Indicates better-than-expected performance after adjusting for patient risk.

W Score is limited

• The W Score method is derived from the MTOS study, which was undergirded by linear methods
• Divides patients into bins of equal width based on predicted survival probability, \(P(Survival)\).
  
• Assumes that \(P(Survival)\) is evenly distributed.

W Score is limited

• Problem: \(P(Survival)\) from logistic regression is not normally distributed — many patients cluster near very high or very low survival probabilities.
  
• Linear bins overrepresent some risk groups and underrepresent others, which can distort observed vs expected comparisons.

Distribution of Predicted Survival

Empirical data show that trauma patients are not evenly distributed across predicted survival probabilities.

Most patients presenting to trauma centers have a very high likelihood of survival.

MTOS Distribution

W-Score Can Be Misleading

• The W-score is heavily influenced by the majority of patients with very high \(P(Survival)\) values (for example, \(P(Survival) > 0.8\).
  
• Because most trauma patients are expected to survive, the W-score often reflects performance among the least acute patients, not those at highest risk.

W-Score Can Be Misleading

• This means two centers could have identical W-scores even if one performs much better with severely injured patients.

Assumption vs. Reality

• The W-score assumes that \(P(Survival)\) values are linearly distributed among patients across the 0–1 range.
  
• However, observed data show that \(P(Survival)\) is highly skewed, with most patients near 1.0.
  
• Therefore, linear bins or evenly spaced \(P(Survival)\) categories overweight low-acuity patients and underweight critical cases.

Take-Home Message on the W Score

• W-score alone provides a partial picture of trauma center performance.
  
• For a fair comparison, models such as the Relative Mortality Metric (RMM) use non-linear binning that reflects the true, non-normal \(P(Survival)\) distribution observed in real trauma data.

Non-Linear Binning: Why It Matters

• Because \(P(Survival)\) is skewed, non-linear bins capture the distribution more accurately.
  
• Examples of non-linear binning:
  • Quantiles (equal number of patients per bin)
    
  • Clinically meaningful thresholds (e.g., very high risk vs moderate vs low)

Non-Linear Binning: Why It Matters

• This allows fairer comparison of observed vs expected outcomes across risk groups.
  
• Ensures that the benchmarking metrics (RMM, W-score) reflect actual patient risk rather than arbitrary binning.

Relative Mortality Metric (RMM): Bin-Based Approach

• RMM compares observed vs. expected mortality while accounting for patient risk distribution.
  
• Patients are grouped into bins based on predicted survival probabilities \(P(Survival)\).
  
• Each bin contributes proportionally to the metric based on its width or patient count.

Relative Mortality Metric (RMM): Bin-Based Approach

\[ \text{RMM} = \frac{\sum_{b=1}^{j} R_b \, (A_b - O_b)}{\sum_{b=1}^{j} R_b \, A_b} \]

Relative Mortality Metric (RMM): Bin-Based Approach

Where:

• \(b = 1, \dots, j\) : bin index, from first to last bin
  
• \(j\) : total number of bins
  
• \(R_b\) : width of bin \(b\) or number of patients in the bin
  
• \(A_b\) : predicted (expected) deaths in bin \(b\)
  
• \(O_b\) : observed deaths in bin \(b\)

Relative Mortality Metric (RMM): Bin-Based Approach

Interpretation for clinicians:
- Positive RMM -> observed mortality is lower than expected, better performance.
- Negative RMM -> observed mortality is higher than expected, worse performance.
- Weighted binning ensures fair comparison across different patient risk levels.
- Easy interpretation on a scale from -1 (bad) to 1 (great), where 0 is “met expectations”.

Why Bin Weighting Matters

• Predicted survival probabilities \(P(Survival)\) are not evenly distributed — most patients may cluster at high or low survival.
  
• Using weighted bins ensures that each risk group contributes appropriately to the RMM.
  
• This prevents over- or under-representation of patient subgroups in the metric.
  
• RMM thus provides a clinically meaningful, risk-adjusted benchmark for trauma center performance.

Key Takeaways for Clinicians on RMM

• RMM and W-score are risk-adjusted metrics, accounting for patient severity and demographics.
  
• M-score linear binning can be misleading because predicted survival probabilities are skewed.
  
• Non-linear binning improves interpretation, particularly for observed vs expected mortality analyses.

Key Takeaways for Clinicians on RMM

• Using these methods allows trauma centers to compare performance fairly and identify opportunities for improvement.

Digging deeper

RMM by Trauma Type

RMM by Age Group

RMM by Biological Sex